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Models for Supersymmetric Quantum Mechanics Aust. J. Phys., 1991, 44, 353-62 Models for Supersymmetric Quantum Mechanics M. Baake,A R. Delbourgo and P. D. Jarvis Department of Physics, University of Tasmania, G.P.O. Box 252C, Hobart, Tas. 7005, Australia. A Institut fur Theoretische Physik, Universitat Tubingen, 7400 Tubingen I, FRG. Abstract An algebraic view of supersymmetric quantum mechanics is taken, emphasising the couplings between bosonic and fermionic modes in the supercharges. A class of model Hamiltonians is introduced wherein the fermionic (bosonic) operators are canonical and the bosonic (fermionic) ones satisfy a Lie algebra (superalgebra) whose representation theory permits the complete solution of the model in principle. The kinematical symmetry of such models is also described. The examples of one and two bosonic models, with 5U(2) and 5U(3) dynamical algebras respectively, are analysed in detail. 1. Introduction and Main Results Supersymmetry provides a unified framework for the exact solutions of classical textbook nonrelativistic quantum mechanical models (Haymaker and Rau 1986; Fuchs 1986). In the simplest version the odd and even parts of the Hamiltonian correspond to 'partner' potentials for the system, whose form is related by supersymmetry and which produce strongly constrained spectra, phase shifts, etc.; in soluble cases the partners are typically reparametrisations of each other. More generally, one can consider conditions under which a wider class of Hamiltonians with a given number of bosonic and fermionic canonical coordinates and momenta possesses simple or extended supersymmetry (de Crombrugghe and Rittenberg 1983; Rittenberg and Yankielowicz 1985), allowing also for constraints (Flume 1985). Making reasonable assumptions about the couplings in the Hamiltonian and supercharges, there are only a limited number of solutions (up to N = 8); the more extended models often require exotic values of magnetic moments, monopole charges and so on, conspiring with highly specific spatial potentials (D'Hoker and Vinet 1985). In the present paper we point out a general class of N = 1 supersymmetric Hamiltonians H = {Q, Qt} which arise from supercharges involving couplings between fermionic and bosonic modes; one set of modes satisfies canonical commutation (or anticommutation) relations while the other set is assumed to satisfy certain Lie algebra (or superalgebra) relations. Primary amongst these is the requirement that they should mutually commute (or anticommute) in order to secure Q2 = Qt2 = O. Further conditions will dictate the type of kinematical and dynamical symmetry possessed by the model Hamiltonian. 0004-9506/91/040353$05.00 354 M. Baake et al. In the next section we treat the bosonic compact case of one canonical mode (n = 1), which produces an 5U(2) dynamical algebra. * When n = 2 the bosonic compact case leads to an SU(3) dynamical algebra; this is considered in detail in the following section as an example of this class of model. Concluding remarks and generalisations are drawn in Section 4. 2. An SU(2) Example As a simple illustration of the construction, consider the n = 1 case consisting of one bosonic and one fermionic operator: Q=Ea, Qt =Fat, where {a,at }= 1, {a,a}={at,at}=O, and F is the hermitian conjugate of E. The requirement Q2 = Qt2 = ° is now trivially satisfied and H is fixed as H = EF - [E,F]NF, where NF = a t a is the number operator for fermionic modes. If we now consider E and F as generators of a Lie algebra, then possible supersymmetric H are enumerated by giving the representation of the Lie algebra and the matrix elements of the various operators therein. The simplest possibility is that E and F generate the angular momentum SU(2): [E,F] = 21o, Uo,E] =E, Uo,F] =-F. Introducing now the orthonormal basis states U m} with m = -j,-j+ 1, ... ,j-1,j; j = 0, 1/2, 1, ... and the well-known matrix elements 1-U m}= ~U+m)U-m + 1) U m -l}, we obtain the spectrum of bosonic (nF = 0) and fermionic (nF = 1) sectors, E5~ = U + m)U - m + 1), E5~ = U - m)U + m + 1). In each sector this corresponds to a singlet E = ° state (m = -j and m = +j, respectively) and a tower of doubly degenerate E > ° states ranging up to Emax=jU+1) for m=O,l and m=O,-l, respectively. 3. Case Study of n = 2 Turning now to the n = 2 bosonic case we consider the Lie algebra generated by two commuting operators El,E2 and their hermitian conjugates Fl,F2. For example if [Fl,E2] oc El, then the Lie algebra is SU(3) or a non-compact real form thereof. In principle any rank 2 semi-simple Lie algebra is admissible, which then becomes the dynamical or spectrum-generating symmetry for the • The non-compact 5U(1,I) case is mentioned in the Conclusions. Supersymmetric Quantum Mechanics 355 supercharges Q=E1al +E2a2' Qt =Fla1 +F2a2 and their associated supersymmetric Hamiltonian; al,a2 and their conjugates aI, a 2 stand for fermionic creation and annihilation operators. In the general bosonic case we would have t _, t Q= ,LEocaoc, Q - LE_()(a()(, oc ()( where the E()( are a set of mutually commuting positive root vectors of the dynamical algebra Land a()(,at are the corresponding set of canonical fermionic operators. The representations of L together with the fermionic Fock space provide the physical Hilbert space of the model; H is in the enveloping algebra of L In addition, there normally is an identifiable kinematical symmetry Xc: L which commutes with H and accounts for some of the degeneracies of the spectrum, apart from the usual supersymmetric doubling of states (Haymaker and Rau 1986). For models generalising Q, Q t above, L is SU(n) and X is SU(n -1); other choices of H lead to different X's. It is our contention that, in contrast to the highly constrained nature of the supersymmetric systems alluded to above, the present type of model may well be realised in simple physical situations, given the simple ansatz taken and the minimal structural assumptions about the bosonic sector. There is thus the hope that models of this type may have a utility akin, say, to the use of group theory for two-level systems, namely SU(2) realised via the Pauli matrices (Dirac 1936). To this extent, the proposal is intermediate between the highly constrained models and models based on explicit supergroup chains, as applied to nuclear spectra (for a review of other applications see Kostelecky and Campbell 1985; Balantekin 1985; Sukumar 1985). We now turn to a detailed examination of the SU(3) case, for both matrix and oscillator realisations. Fixed Irreducible Representation of SU(3) For ease of notation we introduce the entire set of SU(3) generators in matrix form Ej, 1 ~ i,j ~ 3, i k k i i k [Ej,E.el = 8j E.e - 8.e Ej, (1) where I.r=l Ei = 0 and E{ = (Ej)t in unitary representations. We may choose El and E2 as any suitable pair of mutually commuting generators; for example, El == Et E2 == Et and put a == aI, b == a2 so Q== aEl +bE2 and H=={Q,Qt}. (2) Upon using the SU(3) algebra (1), we get H = EIE~ + EfE~ + (1- Na)(E~ - E~) + (1- Nb)(E~ - E~) + a t bE~ + b t aE~, (3) 356 M. Baake et al. where Na and Nb are number operators for fermionic modes of type a and b, respectively. In order to describe states in representations of 5U(3) so as to choose a symmetry-adapted basis. for H, we adopt the Gel'fand labelling scheme (Biedenharn and Louck 1981) p p' 0) q q' r where the rows are ordered by lexicality and successive rows are subject to 'betweenness' conditions, p ~ p' ~ 0, q ~ q' ~ 0, r ~ 0, p ~ q ~ p', p' ~ q' ~ 0, q ~ r ~ q', for the integer labels p,p',q,q' and r; the pair {p,p'} labels the highest weight of the representation corresponding to the partition of two parts of length p and p'. A generic state then has weight given by pp'O pp'O E} I q q' )== [r-(p+p')/3] q q' ), r r p p' 0) p p' 0) E~I q q' ==[q+q'-r-(p+p')/3] q q' , (4) r r PP'O) PP'O) E~ I q q' == [2(p + p')/3 - q - q'] q q' . r r Matrix elements of other E) are then prescribed; for instance (Biedenharn and Louck 1981) p p' p p' ~ ( + 1 °' E1 ,0) == (p - q)(q - p' + 1)(q + 2)(r - q, + 1) . (5) q q 3 q q (q _ q' + 1)(q - q' + 2) r+ 1 r From (2) and (3) the Hamiltonian H acts on the tensor product :F ® V{P,P'} of the 4 dimensional fermionic Fock space :J with the bosonic carrier space Y of the 5U(3) representation with highest weight {p,P'}. Diagonalisation can be carried out explicitly using (5) and similar matrix elements; however the analysis is simplified by identifying which operators provide the kinematical or degeneracy symmetry of the problem by commuting with H. From the form of H it is clear that E~ and NF == Na+Nb are two such operators. Moreover, Q itself can be regarded as a scalar under transformations of the 2,3 indices if there are also contragedient transformations of a,b. To this end, Supersymmetric Quantum Mechanics 357 we introduce the V-spin generators (Lichtenberg 1978) V+=E~, V_=E~, V3 = (E~ - E~)/2 (6) and their fermionic counterparts' v+ =-bt a , v_=-atb, V3 = (bt b-at a)/2, (7) form the total V-spin operators 11±=V±+v±, 113 =V3+ V3 (8) and confirm that [Y,Q] = [Y,Qt] = 0, (9) where 11± == 111 ±i112 as usual.
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